THE CONICS, BOOK V
163
if P' be any other point on it, P'g diminishes as P' moves
farther from P on either side to B or B\ and
Prf — P'g 2 = nn
/2 p'-BB'
BB'
or nn
GA 2 —CB 2
cW
If 0 be any point on Pg produced beyond the minor axis, PO
is the maximum straight line from 0 to the same part of the
ellipse for which Pg is a maximum, i.e. the semi-ellipse BPS',
&c. (Y. 20-2).
In V, 23 it is proved that, if g is on the minor axis, and gP
a maximum straight line to the curve, and if Pg meets A A'
in G, then GP is the minimum straight line from G to the
curve; this is proved by similar triangles. Only one normal
can be drawn from any one point on a conic (V. 24-6). The
normal at any point P of a conic, whether regarded as a
minimum straight line from G on the major axis or (in the
case of the ellipse) as a maximum straight line from g on the
minor axis, is perpendicular to the tangent at P (V. 27-30);
in general (1) if 0 be any point within a conic, and OP be
a maximum or a minimum straight line from 0 to the conic,
the straight line through P perpendicular to PO touches the
conic, and (2) if 0' be any point on OP produced outside the
conic, O'P is the minimum straight line from 0' to the conic,
&c. (V. 31-4).
Number of normals from a point.
We now come to propositions about two or more normals
meeting at a point. If the normal at P meet the axis of
a parabola or the axis A A' of a hyperbola or ellipse in G, the
angle PGA increases as P or G moves farther away from A,
but in the case of the hyperbola the angle will always be less
than the complement of half the angle between the asymptotes.
Two normals at points on the same side of AA' will meet on
the opposite side of that axis; and two normals at points on
the same quadrant of an ellipse a,s A B will meet at a point
within the angle ACB' (V. 35-40). In a parabola or an
ellipse any normal PG will meet the curve again; in the
hyperbola, (1) if A A' be not greater than p, no normal can
meet the curve at a second point on the same branch, but
M 2
